229 under discussion. For example, if it be wished to investigate the properties of a conic section with respect to a pair of tangents and their chord of contact, we write its equation uv + w~ = Q, where ii = 0, t> = 0, w = represent the two tangents and the chord of contact respectively. This procedure has two great advantages. It enables us to greatly abridge the necessary analytical equations, to arrive at them more easily, and thus to lighten or altogether avoid the cumbersome algebraical calculations which had broken the back of the old-fashioned Cartesian geometry and arrested its progress altogether ; and it greatly facilitates the geometrical interpretation of analytical results whether intermediate or final. In the first volume of the Entwickelungen, Pliicker applied the method of abridged notation to the straight line, circle, and conic sections, and he subsequently used it with great effect in many of his researches, notably in his theory of cubic curves. In the second volume of the Entwickelungen, Pliicker clearly established on a firm and independent basis the great principle of duality. This principle had originally been established by Poncelet as a corollary on the theory of the pole and polar of a conic section. Gergonne maintained the independent and fundamental nature of the principle, and hence arose a violent discussion between him and Poncelet into which Pliicker was drawn. He settled the matter in Gergonne s favour by introducing the notion of the coordinates of a line and of a plane, and showing that in plane geometry, for example, wo could with equal readiness represent a point either by means of coordinates or by means of an equation, and that the same was true of a line. Hence it appeared that the point or the line in plane geometry, and the point or the plane in solid geometry, could with equal readiness and with equal reason be taken as elements. It was thus made evident that any system of equations proving a theorem regarding points and lines or regarding points and planes could at once be read as proving another in which the words point and line or the words point and plane were everywhere interchanged. Another subject of importance which Pliicker took up in the Ent wickelungen was the curious paradox noticed by Euler and Cramer, that, when a certain number of the intersections of two algebraical curves are given, the rest are thereby determined. Gergonne had shown that when a number of the intersections of two curves of the (P + (?)th degree lie on a curve of the ^ th degree the rest lie on a curve of the qih degree. Pliicker finally (Gergonne Ann., 1828-29) showed how many points must be taken on a curve of any degree so that curves of the same degree (infinite in number) may be drawn through them, and proved that all the points, beyond the given ones, in which these curves intersect the given one are fixed by the original choice. Later, simultaneously with Jacobi, he extended these results to curves and surfaces of unequal order. Allied to the matter just mentioned was Pliicker s discovery of the six equations connecting the numbers of singularities in algebraical curves. It will be best described in the words of Clebsch : " Cramer was the first to give a more exact discussion of the singularities of alge braical curves. The consideration of singularities in the modern geometrical sense originated with Poncelet. He showed that the class k of a curve of the nth order, which Gergonne by an extra ordinary mistake had considered to be identical with its order, is in general n(n-l); and hence arose a paradox whose explanation became possible only through the theory of the simple singularities. By the principle of duality the order n of a curve should be derived in the same way from the class k as k is from n. But if we derive n in this way from k we return not to n but to a much greater number. Hence there must be causes which effect a reduction during this operation. Poncelet had already recognized that a double point reduces the class by 2, a cusp at least by 3, and a multiple point of the pi order, all of whose tangents are distinct, by p(p-). Here it was that Pliicker took up the question. By first directly determining the number of the points of inflexion, considering the influence of double points and cusps, and finally applying the principle of duality to the result obtained, he was led to the famous formula} for the singularities of curves which bear his name, and which completely resolve the paradox of Poncelet formula} which already in the year 1854 Steiner could cite us the well known, without, however, in any way mentioning Pliicker s name in con nexion with them. Pliicker communicated his formulie in the first place to Cralles Journal, vol xii. (1834), and gave a further exten sion and complete account of his theory in his Thcoric dor Algc- braischen Curvcn, 1839." In 1 833 Pliicker left Bonn for Berlin, where be occupied for a short time a post in the Friedrich Wilhelm s Gymnasium. He was then called in 1834 as ordinary professor of mathe matics to Halle. While there he published his System der Analytischen Geometric, aufneue Betrachtungsweisen gegriin- det, und insbesondere eine Ausfuhrliche Theorie der Curven drifter Ordnung enthaltend, Berlin, 1835. In this work he introduced the use of linear functions in place of the ordinary coordinates, and thereby increased the generality and elegance of his equations ; he also "made the fullest use of the principles of collineation and reciprocity. In fact he develops and applies to plane curves, mainly of the third degree, the methods which he had indicated in the Entwickelungen and in various memoirs published in the interim. His discussion of curves of the third order turned mainly on the nature of their asymptotes, and depended on the fact that the equation to every such curve can be put into the form pqr + /AS = 0. He gives a com plete enumeration of them, including two hundred and nineteen species. In 1836 Pliicker returned to Bonn as ordinary professor of mathematics. Here he published his Theorie der Algebraischen Curven which formed a con tinuation of the System der Analytischen Geometric. The work falls into two parts, which treat of the asymptotes and singularities of algebraical curves respectively ; and extensive use is made of the method of counting constants which plays so large a part in modern geometrical researches. Among the results given we may mention the enumeration of curves of the fourth order according to the nature of their asymptotes, and according to the nature of their singularities, and the determination for the first time of the number of double tangents of a curve of the fourth order devoid of singular points. From this time Pliicker s geometrical researches practi cally ceased, only to be resumed towards the end of his life. It is true that he published in 1846 his System der Geometric des Kaumes in neuer Analytischer Behandlunys- iveise, but this contains merely a more systematic and polished rendering of his earlier results. It has been said that this cessation from pure mathematical work was due to the inappreciative reception accorded by his country men to his labours, and to their jealousy of his fame in other lands ; it seems likely, however, that it was due in some degree to the fact that he was called upon to under take the work of the physical chair at Bonn in addition to his proper duties. In 1847 he was made actual professor of physics, and from that time his wondrous scientific activity took a new and astonishing turn. Pliicker now devoted himself to experimental physics in the strictest sense as exclusively as he had formerly done to pure mathematics, and with equally brilliant results. His first physical memoir, published in Poggendor/ s Annalen, vol. Ixxii., 1847, contains his great discovery of magnecrystallic action. Then followed a long series of researches, mostly published in the same journal, on the properties of magnetic and diamagnetic bodies, establishing results which are now part and parcel of our magnetic knowledge. It is unnecessary here to analyse these re searches, of which an account has been given in the article MAGNETISM (vol. xv. p. 262 sq.) ; it will be sufficient to say that in this work Pliicker was the worthy collaborateur, and, had it not been that their fast friendship and mutual admiration renders the word inappropriate, we might have said rival, of Faraday. In 1858 (Pogg. Ann., vol. ciii.) he published the first of his classical researches on the action of the magnet on the electric discharge in rarefied gases (see ELECTRICITY, vol. viii. p. 74). It is needless now to dilate upon the great beauty and importance of these researches, which remain the leading lights in one of the darkest channels of magnetic science. All the best work that has recently been done on this important subject is simply development of what Pliicker did, and in some instances (notably in many of the researches of Crookes) merely reproduction on a larger scale of his results. Pliicker, first by himself and afterwards in conjunction with Hittorf, made many important discoveries in the spectroscopy of gases. He was the first to use the vacuum
tube with the capillary part now called a Geissler s tube,